PHYSICAL REVIEW B 83, 174441 (2011) Electronic and magnetic properties of triangular graphene quantum rings P. Potasz, 1,2 A. D. G ¨ uc ¸l¨ u, 1 O. Voznyy, 1 J. A. Folk, 3 and P. Hawrylak 1 1 Institute for Microstructural Sciences, National Research Council of Canada, Ottawa, Canada 2 Institute of Physics, Wroclaw University of Technology, Wroclaw, Poland 3 Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia, Canada (Received 9 February 2011; revised manuscript received 6 April 2011; published 31 May 2011) Electronic and magnetic properties of triangular graphene rings potentially fabricated using carbon nanotubes as masks are described as a function of their size and width. The electronic properties of the charge neutral system are calculated using tight-binding method and interactions are treated using the mean-field Hubbard model. We show that for triangular quantum dots with a triangular hole, the magnetic properties are determined by the width of the ring, leading to ferromagnetic corners and antiferrimagnetic sides. The electronic properties of gated graphene quantum rings as a function of additional number of electrons or holes are described by a combination of tight-binding, Hartree-Fock, and configuration interaction methods. The outer edge is found to be maximally spin polarized for almost all filling factors while the evolution of the excitation gap as a function of shell filling shows oscillations as a result of electronic correlations. DOI: 10.1103/PhysRevB.83.174441 PACS number(s): 75.75.−c, 73.22.Pr, 81.05.ue I. INTRODUCTION There is currently significant interest in electronic properties 1–5 and potential technological applications of graphene. 5–8 While ideal carbon graphene sheet is in itself nonmagnetic, theoretical research suggests occurrence of local magnetic moments in the vicinity of defects 9,10 or zigzag-type boundaries of graphene sheet. 11–15 The zigzag edges lead to the existence of degenerate states near the Fermi level, predicted by tight-binding model and confirmed by density functional theory (DFT) calculations. 11,16–21 Experimentally, these states were observed using scanning tunneling microscope (STM) near monoatomic steps on a graphite surface. 22,23 For graphene nanostructures with zigzag edges, electron exchange interactions lead to antiferrimagnetic order in graphene ribbons 11 but ferromagnetic order in triangular graphene quantum dots (TGQD). 13,14,20,24 Apart from edge magnetism, interactions in partially filled edge states result in other correlated ground states. 24,25 In particular, magnetism in TGQD can be completely destroyed by adding extra electron to the charge neutral system. 24 The fabrication of triangular graphene quantum dots with well-defined shape and zigzag edges is potentially challenging as it requires control of the edges at the atomic level. In this work we explore the use of carbon nanotubes (CNTs) as masks with atomically precise shape and size for producing graphene triangular nanostructures. As shown in Fig. 1 three carbon nanotubes allow the fabrication of a triangular graphene quantum dot but with a hollow center, triangular graphene quantum ring (TGQR). In this paper, we describe electronic and magnetic properties of TGQR as a function of size, width and charge controlled by the gate. We show that many of the properties of triangular graphene quantum dots survive and that new properties related to ring formation appear. The paper is organized as follows. In Sec. II, we de- scribe our triangular graphene ring model and propose an experimental method for designing graphene nanostructures with well-defined edges. In Sec. III, we study the single particle properties using tight-binding (TB) Hamiltonian in the nearest neighbor approximation. In Sec. IV, we investigate magnetic properties of the charge neutral rings using mean- field Hubbard model and density functional theory (DFT). In Sec. V, we study magnetic properties and the role of correlations in gated charged ring using a combination of tight- binding, Hartree-Fock, and configuration interaction methods (TB + HF + CI). We analyze electronic and magnetic proper- ties of graphene rings with given width as a function of the number of electrons occupying degenerate shell. Section VI contains the summary. II. TRIANGULAR GRAPHENE QUANTUM RING MODEL As shown in Fig. 1, TGQR can be fabricated using carbon nanotubes (CNT) as a mask in the etching process. 26,27 One can place CNT over the graphene sheet along a given crystallographic direction and cover atoms lying below, e.g., along a zigzag direction. Three carbon nanotubes can be arranged in a triangular shape, along three zigzag edges. As a result one expects to obtain a triangular structure with well defined zigzag edges and a hole in the center, as shown on the right in Fig. 1. We also note that this method can be used to obtain, e.g., bow-tie structures with potential application as quantum information logic devices. 14 TGQD with a zigzag edge shown in Fig. 1 consists of N 2 out + 4N out + 1 atoms, 28 where N out is the number of edge atoms on one outer edge and edge atoms are defined as those having only two neighbors. In order to create a triangular ring, we remove a small triangle from the center (Fig. 1). The small triangle consists of N 2 inn + 4N inn + 1 atoms, where N inn is the number of edge atoms on one inner edge. The resulting TGQR has N = N 2 out − N 2 inn + 4(N out − N inn ) atoms. Its width satisfies N out − N inn = 3(N width + 1), where N width is the width counted in the number of benzene rings. For instance, the structure in Fig. 1 has N width = 2. In the full triangle, the imbalance between number of A type (N A ) and B type (N B ) of atoms in bipartite honeycomb graphene lattice, proportional to N out , leads to appearance of zero-energy states in the TB model in the nearest-neighbors approximation. The number of zero-energy states can be defined as N zero =|N A − N B |. 29 Removing a small triangle from the center lowers the imbalance between 174441-1 1098-0121/2011/83(17)/174441(6) Published by the American Physical Society